I noticed after first play in MP (when I too could not use any powers) - make sure your points are placed into your powers (like leveling up) through "Character." they will then show with the corresponding buttons at the bottom of your screen while playing
In Multiplayer there is no power-wheel and weapon-wheel present unlike in single-player. Instead you have three buttons assigned to every power that your particular character can utilize; Y, LB, and RB.
If you find that nothing happens when pressing any of those then you haven't actually allocated any points to a power after you've deployed the character, and before you accept the mission when you're in the lobby screen. Such as seen
here as an example.
Because a point is allocated into the Biotic Charge power slot, as seen in the picture, it becomes available to use for whatever respected button it's assigned to automatically.
For every different class/character you utilize and deploy in the cooperative mode, Soldier, Engineer, Adept, Sentinel, Vanguard, and Infiltrator, you will have to level up the abilities before hand to use any powers, just like in single-player.